existence and nonexistence results for critical growth

Calculus of variations manuscript No.(will be inserted by the editor)

Filippo Gazzola ·Hans–Christoph Grunau ·Marco Squassina

Existence and nonexistence results for criticalgrowth biharmonic elliptic equations

Received: date / Accepted: date /Published online: date – c© Springer-Verlag 2002

Abstract. We prove existence of nontrivial solutions to semilinear fourth orderproblems at critical growth in some contractible domains which are perturba-tions of small capacity of domains having nontrivial topology. Compared withthe second order case, some difficulties arise which are overcome by a decomposi-tion method with respect to pairs of dual cones. In the case of Navier boundaryconditions, further technical problems have to be solved by means of a carefulapplication of concentration compactness lemmas. The required generalization ofa Struwe type compactness lemma needs a somehow involved discussion of certainlimit procedures.

Also nonexistence results for positive solutions in the ball are obtained, ex-tending a result of Pucci and Serrin on so-called critical dimensions to Navierboundary conditions. A Sobolev inequality with optimal constant and remainderterm is proved, which is closely related to the critical dimension phenomenon.Here, this inequality serves as a tool in the proof of the existence results and inparticular in the discussion of certain relevant energy levels.

Mathematics Subject Classification (2000): 35J65; 35J40, 58E05

1. Introduction

We consider the following biharmonic critical growth problem

∆2u = λu+ |u|2∗−2u in Ω (1)

either with Navier boundary conditions

u = ∆u = 0 on ∂Ω (2)

F. Gazzola: Dipartimento di Scienze e Tecnologie Avanzate,Corso Borsalino 54, I–15100 Alessandria, Italy. e-mail: [email protected]

H.–Ch. Grunau: Fakultat fur Mathematik der Universitat, Postfach 4120,D–39016 Magdeburg, Germany. e-mail: [email protected]

M. Squassina: Dipartimento di Matematica e FisicaVia Musei 41, I–25121 Brescia, Italy. e-mail: [email protected]

The first author was supported by MURST project “Metodi Variazionali edEquazioni Differenziali non Lineari”

2 Filippo Gazzola et al.

or with Dirichlet boundary conditions

u = ∇u = 0 on ∂Ω. (3)

Here Ω ⊂ Rn (n ≥ 5) is a bounded smooth domain, λ ≥ 0 and 2∗ = 2nn−4 is

the critical Sobolev exponent for the embedding H2(Rn) → L2∗(Rn). Weare interested in existence and nonexistence of nontrivial solutions of (1)with one of the above boundary conditions.

We consider first the case where λ = 0, namely the equation

∆2u = |u|8/(n−4)u in Ω. (4)

It is well known that (4)–(2) and (4)–(3) admit no positive solutions ifΩ is star shaped (see [27, Theorem 3.3] and [30, Corollary 1]). Clearly,these are not exhaustive nonexistence results as they only deal with posi-tive solutions, see also Section 3 below for further comments. On the otherhand, by combining the Pohozaev identity [34] with the unique continuationproperty [22, Section 3], it is known that one indeed has nonexistence re-sults of any nontrivial solution for the correponding second order equation−∆u = |u|4/(n−2)u in star shaped domains. This suggests that, in order toobtain existence results for (4), one should either add subcritical perturba-tions like the linear term λu in (1) or modify the topology or the geometryof the domain Ω. For general subcritical perturbations we refer to [4,13] andreferences therein. Domains with nontrivial topology are studied in [3] forDirichlet and in [11] for Navier boundary conditions. Here we are interestedin topologically simple but geometrically complicated domains. As far as weare aware, the case of “strange” contractible domains has not been tackledbefore and this is precisely the first aim of the present paper. We show that(4)–(2) admits a positive solution in a suitable contractible domain Ω. Thisresult is the exact counterpart of [27, Theorem 3.3] where it is shown that(4)–(2) admits no positive solution if Ω is star shaped. For the Dirichletboundary condition we obtain a weaker result: we show that (4)–(3) admitsa nontrivial solution in the same kind of contractible domain. Clearly, thisstill leaves open some problems concerning Dirichlet boundary conditions,see Section 3. On one hand our proof is inspired by strong arguments de-veloped by Passaseo [32], on the other hand we have to face several harddifficulties, especially (and somehow unexpectedly) under Navier boundaryconditions. One of the crucial steps in the approach by Passaseo is to provethat sign changing solutions of (4) “double the energy” of the associatedfunctional. For the second order problem this may be shown by the usualtechnique of testing the equation with the positive and negative parts of thesolution. Of course, this technique fails for (4) where higher order derivativesare involved and we overcome this difficulty thanks to the decompositionmethod in dual cones developed in [14]. This method enables us to bypassthe lack of nonexistence results for nodal solutions of (4) in star shapeddomains. Moreover, when dealing with Navier boundary conditions, the re-quired generalization of the Struwe compactness lemma [37] turns out to

Biharmonic equations at critical growth 3

be very delicate because of the second boundary datum and the lack of auniform extension operator for H2 ∩ H1

0–functions in families of domains.See Lemma 5 and its proof in Section 7. The same problem arises in Lemma7 where a uniform lower bound for an enlarged optimal Sobolev constanthas to be found in a suitable class of domains. An important tool for thisLemma is a Sobolev inequality with optimal constant and remainder term,see Theorem 5. This inequality is closely related to nonexistence results,which will be summarized in what follows.

Next, we restrict our attention to the case where λ > 0 and Ω = B, theunit ball in Rn. We are interested in positive radially symmetric solutions.Let λ1 > 0 be the first eigenvalue of ∆2 with homogeneous Dirichlet bound-ary conditions. A celebrated result by Pucci–Serrin [36] shows that (1)–(3)admits a nontrivial radially symmetric solution for all λ ∈ (0, λ1) if andonly if n ≥ 8 (n is the space dimension). If n = 5, 6, 7 the range 0 < λ < λ1

is no longer correct since there exists λ∗ ∈ (0, λ1) such that (1)–(3) admitsno nontrivial radial solution whenever λ ∈ (0, λ∗]; moreover, there existsλ∗ ∈ [λ∗, λ1) (presumably λ∗ = λ∗) such that (1)–(3) admits even a pos-itive radially symmetric solution for all λ ∈ (λ∗, λ1). In other words, thewell known [6] nonexistence result for radially symmetric solutions of thesecond order problem for small λ in dimension n = 3 carries over to (1)–(3)in dimensions n = 5, 6, 7: Pucci–Serrin call these dimensions critical. Fur-thermore, the space dimensions n for which one has nonexistence of positiveradially symmetric solutions of (1)–(3) for λ in some (right) neighbourhoodof 0 are called weakly critical [17]. In Theorem 3 below we prove that thedimensions n = 5, 6, 7 are weakly critical also for Navier boundary condi-tions. Therefore, critical dimensions seem not to depend on the boundaryconditions. This result is perhaps related to the fact that critical dimensionsmay have an explanation in terms of the summability of the fundamentalsolution corresponding to the differential operator ∆2, see [20,26].

Finally, when Ω = B we show that (4)–(3) admits no nontrivial radiallysymmetric solution; note that no positivity assumptions on the solution aremade. This result was already obtained in [16, Theorem 3.11]; we give herea different (and simpler) proof.

2. Existence results

By D2,2(Rn) we denote the completion of the space C∞c (Rn) with respect

to the norm ‖f‖22,2 =∫

Rn |∆f |2; this is a Hilbert space when endowed withthe scalar product

(f, g)2,2 =∫

Rn

∆f∆g dx.

By solution of (1) we mean here a function u ∈ H2 ∩H10 (Ω) (when dealing

with (2)) or a function u ∈ H20 (Ω) (when dealing with (3)) which satisfies∫

Ω

∆u∆ϕdx =∫

Ω

(λuϕ+ |u|8/(n−4)uϕ

)dx


for all ϕ ∈ H2 ∩H10 (Ω) (resp. H2

0 (Ω)). For Dirichlet boundary conditionsthis variational formulation is standard, while for Navier boundary datawe refer to [4, p.221]. By well known regularity results (see [25, Theorem1] and [43, Lemma B3]), if ∂Ω ∈ C4,α, then the solutions u of (1) satisfyu ∈ C4,α(Ω) and solve (1) in the classical sense.

Definition 1. Let K ⊂ Rn bounded. We say that u ∈ D2,2(Rn) satisfiesu ≥ 1 on K in the sense of D2,2(Rn) if there exists a sequence (uh) inC2

c (Rn) such that uh ≥ 1 on K for each h ∈ N and uh → u in D2,2(Rn).Analogously, u ≤ 1 and u = 1 on K are defined.

Definition 2. We define the (2, 2)−capacity of K (capK) as

capK = inf∫

Rn

|∆u|2 dx : u = 1 on K in the D2,2(Rn) sense.

We set cap ∅ := 0.

Since the nonempty setu ∈ D2,2(Rn) : u = 1 on K in the D2,2(Rn) sense

is closed and convex, there exists a unique function zK ∈ D2,2(Rn) suchthat zK = 1 on K and ∫

Rn

|∆zK |2 dx = capK.

Finally we make precise what we mean by set deformations:

Definition 3. Let Ω ⊂ Rn and let H,Ω ⊂ Ω. We say that H can bedeformed in Ω into a subset of Ω if there exists a continuous function

H : H × [0, 1] → Ω

such that H (x, 0) = x and H (x, 1) ∈ Ω for all x ∈ H.

Our first result states the existence of positive solutions for the criti-cal growth equation (4) with Navier boundary conditions. Combined withthe already mentioned nonexistence result in star shaped domains [27], thisshows that the existence of positive solution strongly depends on the geom-etry of the domain.

Theorem 1. Let Ω be a smooth bounded domain of Rn (n ≥ 5) and letH be a closed subset contained in Ω. Then there exists ε > 0 such that ifΩ ⊂ Ω is a smooth domain with cap(Ω \Ω) < ε and such that H cannot bedeformed in Ω into a subset of Ω then there exists a positive solution of

∆2u = u(n+4)/(n−4) in Ωu = ∆u = 0 on ∂Ω.

(5)


We will prove this result in Section 5.Next we turn to Dirichlet boundary conditions. In this case, we merely

show the existence of nontrivial solutions to (4). The lack of informationabout the sign of the solution is due to the lack of information about thesign of the corresponding Green’s function. Moreover, in general domainssign change has even to be expected. See [18] for a survey on this featureand for further references.

Theorem 2. Let Ω be a smooth bounded domain of Rn (n ≥ 5) and letH be a closed subset contained in Ω. Then there exists ε > 0 such that ifΩ ⊂ Ω is a smooth domain with cap(Ω \Ω) < ε and such that H cannot bedeformed in Ω into a subset of Ω then there exists a nontrivial solution of

∆2u = |u|8/(n−4)u in Ωu = ∇u = 0 on ∂Ω.

(6)

The proof of this result is simpler than the one of Theorem 1. As we willexplain below, for (5), one has to study very carefully the behaviour of suit-able sequences uh ∈ H2∩H1

0 (Ωh) for varying domains Ωh. In contrast withthe spaces H2

0 (Ωh), there is no “trivial” extension operator into H2(Rn).Only as positivity of solutions is concerned, the situation with respect toDirichlet boundary conditions is more involved than with respect to Navierboundary conditions. Here we have no positivity statement of the solutionbecause Lemma 3 below does not seem to hold.

If Ω is an annulus and if one restricts to radial functions, the problemis no longer critical and then both (5) and (6) admit a nontrivial solution.For the second order problem −∆u = |u|4/(n−2)u in Ω, u = 0 on ∂Ω, itwas shown with help of very subtle methods by Coron [7] and Bahri–Coron[2] that this result extends to domains with nontrivial topology. Recently,corresponding results have been found also in the higher order case: see [11]for problem (5) and [3] for the polyharmonic analogue of (6) under Dirichletboundary conditions. However, the solutions which are constructed in thesepapers and here are not related. To explain this, assume that Ω satisfiesthe assumptions of [11] or [3]. If uΩ is the solution of (5) or (6), then byexploiting the proof of Theorem 1 one has that uΩ converges weakly tozero as cap(Ω \ Ω) → 0. Hence, any nontrivial solution in Ω may not beobtained as limit of the solutions uΩ in Ω. On the other hand one expectsthe nontrivial solutions in Ω to be stable and to remain under “small”perturbations Ω ⊂ Ω. These solutions in Ω will be different from ours. Thelatter situation was studied in [9] for the second order problem.

To conclude the section, we observe that there are also contractible do-mains Ω, which satisfy the assumptions of the preceding theorems. In thefollowing example we describe precisely such a situation. Further examplesmay be adapted to the biharmonic setting from [32, pp. 39–41].

Example 1. We consider an annular shaped domain as Ω ⊂ Rn with n > 5,where we drill a sufficiently “thin” cylindrical hole along a straight line inorder to obtain the smooth contractible subdomain Ω.


To be more precise: we assume that for ε small enough, Ω \ Ωε is con-tained in a cylinder with basis Bε ⊂ Rn−1 and fixed height. Then by simplescaling arguments one finds that cap

(Ω \Ωε

)= O(εn−5) → 0 as ε → 0,

provided that the dimension satisfies n > 5.We choose H to be a spherical hypersurface in Ω, which cannot be

deformed into a subset of Ωε. This can be seen by looking at the degreeof mapping d(H ( . , t),H, 0) for t ∈ [0, 1], where H is assumed to existaccording to Definition 3.

Instead of the above mentioned segment of fixed length, one may considerany bounded piece of a fixed generalized plane, provided that its codimen-sion is at least 5.

3. Nonexistence results and a Sobolev inequality with optimalconstant and remainder term

We collect here a number of known as well as of new nonexistence resultsfor the equation

∆2u = λu+ |u|8/(n−4)u in Ω (7)

under either Navier or Dirichlet boundary conditions. Here, we include thelinear term λu. It turns out that the discussion of the borderline case λ = 0,where the purely critical equation has to be considered, is somehow involvedand still not exhaustive. This lack of nonexistence results complicates theformulation of the compactness result Lemma 5 below.

The functional analytic counterpart of nonexistence results for (7) forsmall λ > 0 in so-called critical dimensions are Sobolev inequalities withoptimal constant and remainder term. (This becomes clear in the proof ofTheorem 5. See also [14].) Such an inequality in H2

0 (Ω) – i.e. under Dirichletboundary conditions – was proved in [14] while the result in H2 ∩H1

0 (Ω) –i.e. under Navier boundary conditions – is provided in Sect. 3.3 below. Inthe present paper, this refined Sobolev inequality is applied to show thatthe optimal Sobolev constant can be increased uniformly for all subdomainsΩ ⊂ Ω with help of a condition on the “barycenter” of the functions u ∈H2 ∩H1

0 (Ω). See Lemma 7 below.

3.1. Nonexistence under Navier boundary conditions

It turns out that nonexistence questions in this case are relatively hard andthat only restricted results are available.

The case λ = 0In this case, no positive solution to (5) may exist, provided Ω is star shaped.See [42, Theorem 3.10] and also [27, Theorem 3.3]. The key ingredient is aPohozaev type identity [35]. By means of techniques developed by Pucci andSerrin in [36], such a nonexistence result may be also shown for nontrivialradial solutions in the ball.


More difficult seems this nonexistence result for any nontrivial solutionof (5). In this case the Pohozaev type identity is not powerful enough toextend u outside Ω by zero as a solution of (4) in Rn and then to apply theunique continuation property [33].

The case λ > 0In Section 6 we prove the following result.

Theorem 3. Let n ∈ 5, 6, 7 and B ⊂ Rn be the unit ball. Then thereexists a number λ∗ > 0 such that the problem

∆2u = λu+ u(n+4)/(n−4) in Bu = ∆u = 0 on ∂B

(8)

admits no positive solution if λ ∈ (0, λ∗].

By [41], positive solutions to (8) are radially symmetric.Theorem 3 answers a question left open by van der Vorst in [44, Theo-

rem 3]. Moreover, in [44, Theorem 1] complementary existence results wereshown implying that such a nonexistence result can hold at most in thedimensions 5, 6, 7.

These are the critical dimensions for problem (8), at least, when restrict-ing to positive solutions in the ball. Theorem 3 can also be shown just fornontrivial radial solutions: here one has to combine methods of [36] withsome techniques, which appear in the proof of Theorem 4 below.

Again, nonexistence of any nontrivial solution, although expected, hasto be left open.

The case λ < 0In the preceding nonexistence results, the positivity of a solution, the nonex-istence of which had to be shown, is intensively exploited. For λ ≥ 0 onecan conclude from u ≥ 0 that even −∆u ≥ 0 in Ω.

For λ negative, we can only argue as before, when λ is close enough to0, see [28,21]. If λ << 0, the framework of positive solutions is no longeradequate for (7), and the argument in [42, pp. 390/391] and [44, Theorem3] breaks down. Instead, one has to look for nonexistence of any nontrivialsolution. But such a result seems to be still unknown, even for nontrivialradial solutions in the ball.

3.2. Nonexistence under Dirichlet boundary conditions

In this situation, existence results for positive solutions are more involvedthan under Navier boundary conditions (see [15]) and cannot even be ex-pected in general domains. On the other hand, we know much more aboutnonexistence for relatively large classes of solutions.

The case λ < 0Nonexistence of any nontrivial solution is shown in [35], provided the domainΩ is star shaped. This proof is based on a generalized Pohozaev identity


and covers also the general polyharmonic problem with (−∆)K as linearprincipal part.

The case λ > 0In [36] we already find the following result, which gives a stronger statementthan the analogous Theorem 3 above for Navier boundary conditions:

Let n ∈ 5, 6, 7 and assume that Ω is the unit ball B. Then there ex-ists a number λ∗ > 0 such that a necessary condition for a nontrivialradial solution to (6) to exist is λ > λ∗.

As this special behaviour of the critical growth Dirichlet problem (6) maybe observed only in space dimensions 5, 6 and 7, Pucci and Serrin call thesedimensions critical. For the critical dimension phenomenon for general poly-harmonic problems with Dirichlet boundary conditions we refer to [17].

The case λ = 0Also in this case, the nonexistence of any nontrivial solution to (6) has tobe left open, only more restricted results are available. Oswald [30] provedthe nonexistence of positive solutions in strictly star shaped domains bymeans of a Pohozaev type identity (see e.g. [27, (2.6)]) and strong maximumprinciples for superharmonic functions.

Moreover, we can also exclude nontrivial radial solutions:

Theorem 4. Let B ⊂ Rn (n ≥ 5) be the unit ball. Then problem∆2u = |u|8/(n−4)u in B,

u = ∇u = 0 on ∂B(9)

admits no nontrivial radial solution.

This result can be found in [16, Theorem 3.11]. The proof there is basedon a refined integral identity of Pucci and Serrin [36] and some Hardy typeembedding inequalities. In Section 6 we will give a more elementary andgeometric proof.

Unfortunately neither this proof nor the one given in [16, Theorem 3.11]carry over to the general polyharmonic problem, where the discussion ofnonexistence of nontrivial radial solutions in the borderline case λ = 0remains essentially open.

3.3. An optimal Sobolev inequality with remainder term

Sobolev embeddings with critical exponents and the corresponding optimalSobolev constants will play a crucial role in the proof of our existence results.

Let us set

S = infu∈D2,2(Rn)\0

‖∆u‖22‖u‖22∗

. (10)


When working in the whole Rn, this optimal Sobolev constant is at-tained. According to Lemma 2 below, minimizers are necessarily of onesign. Positive solutions of the corresponding Euler Lagrange equations areknown to be of the following form.

Lemma 1. The constant S in (10) is a minimum and (up to nontrivialreal multiples) it is attained only by the functions

uε,x0(x) =[(n− 4)(n− 2)n(n+ 2)](n−4)/8ε(n−4)/2

(ε2 + |x− x0|2)(n−4)/2, (11)

for any x0 ∈ Rn and each ε > 0. Moreover, the functions uε,x0 are the onlypositive solutions of the equation

∆2u = u(n+4)/(n−4) in Rn. (12)

Proof. From Lemma 2 below, we take that any minimizer of (10) may beassumed to be positive. Then, the result follows from [12, Theorem 2.1], [39,Theorem 4] and [23, Theorem 1.3].

Similarly one can define S(Ω) for any Ω ⊂ Rn. It is well known thatS(Ω) is not attained if Ω is bounded [12] and that it does not depend neitheron the domain nor on the boundary conditions [43]. The following extensionof the embedding inequality corresponding to (10) with optimal constant Sgives a quantitative formulation for the fact that S is not attained. Here, itwill be used to show Lemma 7, which in turn is an important step in theproof of our existence results.

Theorem 5. Let Ω ⊂ Rn be open and of finite measure and p ∈ [1, nn−4 ).

Then there exists a constant C = C(n, p, |Ω|) > 0, such that for everyu ∈ H2 ∩H1

0 (Ω), one has∫Ω

|∆u|2 dx ≥ S ‖u‖22∗ +1C‖u‖2p. (13)

Here, |Ω| denotes the n-dimensional Lebesgue measure of Ω. Moreover, ifω > 0, then (13) holds with a constant C = C(n, p, ω) for all Ω with |Ω| ≤ ω.

For the smaller class H20 (Ω) a more refined result can be found in [14].

The proof there is based on an extension and decomposition method (seealso Section 4 below), by means of which analogues to (13) for the spacesHm

0 (Ω) of any order can be also obtained. This proof, however, does notseem to carry over directly to the space H2∩H1

0 due to the lack of informa-tion at the boundary. In Section 6 we sketch a technical proof of Theorem 5,which is based on nonexistence results like in Theorem 3 and on Talenti’ssymmetrization result [40].


4. Preliminary results

In the previous section we have briefly discussed energy minimizing solu-tions of (12). These are necessarily of one sign (this is obtained as a byprod-uct of the following lemma) and of the form given in Lemma 1. The nextresults show that nodal solutions of (4) double the energy level of the corre-sponding “free” energy functional (observe that here, we are working with aconstrained functional). This will allow us to bypass the lack of nonexistenceresults for nodal solutions of (4) in star shaped domains.

Lemma 2. Let u ∈ D2,2(Rn) be a nodal solution of the equation

∆2u = |u|8/(n−4)u in Rn. (14)

Then ‖∆u‖22 ≥ 24/nS‖u‖22∗ .

Proof. Consider the convex closed cone

K =u ∈ D2,2(Rn) : u ≥ 0 a.e. in Rn

,

and its dual cone

K ′ =u ∈ D2,2(Rn) : (u, v)2,2 ≤ 0 ∀v ∈ K

.

Let us show that K ′ ⊆ −K . For each h ∈ C∞c (Rn) ∩ K consider the

solution uh of the problem

∆2uh = h in Rn.

Then by the positivity of the fundamental solution of ∆2 in Rn, it followsuh ∈ K and thus if v ∈ K ′,

∀h ∈ C∞c (Rn) ∩K :

∫Rn

hv dx =∫

Rn

∆2uhv dx = (uh, v)2,2 ≤ 0.

By a density argument one obtains v ≤ 0 a.e. in Rn, i.e. v ∈ −K . Now,by a result of Moreau [29], for each u ∈ D2,2(Rn) there exists a unique pair(u1, u2) in K ×K ′ such that

u = u1 + u2, (u1, u2)2,2 = 0. (15)

Let u be a nodal solution of (14) and let u1 ∈ K and u2 ∈ K ′ be thecomponents of u according to this decomposition. We obtain that ui 6≡ 0and

|u(x)|2∗−2u(x)ui(x) ≤ |ui(x)|2∗ , i = 1, 2, (16)

for a.e. x ∈ Rn. Indeed, if i = 1 and u(x) ≤ 0 then (16) is trivial, while ifu(x) ≥ 0, since u2 ∈ −K one has u(x) = u1(x) + u2(x) ≤ u1(x) and again


(16) holds. The case i = 2 is similar. By combining the Sobolev inequalitywith (15) and (16), we get for i = 1, 2

S‖ui‖22∗ ≤ ‖ui‖22,2 =∫

Rn

∆u∆ui dx =∫

Rn

∆2uui dx =

=∫

Rn

|u|2∗−2uui dx ≤∫

Rn

|ui|2∗ dx = ‖ui‖2∗2∗ ,

which implies ‖ui‖22∗ ≥ S(n−4)/4 for i = 1, 2. Hence, again by (15), oneobtains

‖∆u‖22‖u‖22∗

=[∫

Rn

|∆u|2 dx] [∫

Rn

|u|2∗ dx]−2/2∗

=[∫

Rn

|∆u|2 dx]4/n

=

=[∫

Rn

|∆u1|2 dx+∫

Rn

|∆u2|2 dx]4/n

≥

≥[S‖u1‖22∗ + S‖u2‖22∗

]4/n ≥ 24/nS ,

which concludes the proof.

In a completely similar fashion, one may extend the previous result toany domain in case of Navier boundary conditions. For Dirichlet boundaryconditions this seems not possible due to the lack of information about thepositivity of the corresponding Green’s function.

Lemma 3. Let Ω be a bounded domain of Rn and assume that u solves (5)and changes sign. Then ‖∆u‖22 ≥ 24/nS‖u‖22∗ .

In the case of the half space, by exploiting nonexistence results for pos-itive solutions, we have a stronger result.

Lemma 4. Let Ω = xn > 0 be the half space and assume that u ∈D2,2(Ω) solves the equation

∆2u = |u|8/(n−4)u in xn > 0 (17)

with boundary data either (2) or (3). Then ‖∆u‖22 ≥ 24/nS‖u‖22∗ .

Proof. Notice first that by [27, Theorem 3.3], equation (17)–(2) does notadmit positive solutions. A similar nonexistence result holds for boundaryconditions (3), see [30, Corollary 1] which may be easily extended to un-bounded domains.

Therefore, any nontrivial solution of (17) (with boundary conditions (2)or (3)) necessarily changes sign. We obtain the result if we repeat the ar-gument of the proof of Lemma 2. With Navier boundary conditions this isstraightforward, while with Dirichlet boundary data one may invoke Bog-gio’s principle [5] in the half space.


As pointed out in the introduction, the H2 ∩ H10–framework is more

involved than the H20–case. Therefore, from now on we restrict our attention

to the first situation.Consider the functional f : V (Ω) → R

f(u) =∫

Ω

|∆u|2 dx (18)

constrained on the manifold

V (Ω) =u ∈ H2 ∩H1

0 (Ω) :∫

Ω

|u|2∗ dx = 1. (19)

We say that (uh) ⊂ V (Ω) is a Palais–Smale sequence for f at level c if

limhf(uh) = c, lim

h‖f ′(uh)‖(Tuh

V (Ω))∗ → 0

where TuhV (Ω) denotes the tangent space to the manifold V (Ω) at uh.

We can now state a global compactness result for the biharmonic oper-ator, in the spirit of [37]. Even if the proof is on the lines of that of Struwe,some difficulties arise. In the appendix we give an outline of the proof em-phasizing the main differences with respect to second order equations.

Lemma 5. Let (uh) ⊂ V (Ω) be a Palais–Smale sequence for f at level c ∈R. Then, for a suitable subsequence, one has the following alternative: either(uh) is relatively compact in H2 ∩H1

0 (Ω) or there exist k nonzero functionsu1, ..., uk ∈ D2,2, solving either (14) or (17) with boundary condition (2)and a solution u0 ∈ H2 ∩H1

0 (Ω) of (5) such that

uh u0

k∑j=0

∫|uj |2∗ dx

−1/2∗

in H2 ∩H10 (Ω) (20)

and

limhf(uh) =

k∑j=0

∫|∆uj |2 dx

k∑j=0

∫|uj |2∗ dx

−2/2∗

. (21)

The domain of integration for u0 is just Ω, while for u1, ..., uk, it is eithera half space or the whole Rn.

When working in H20 (Ω) a similar compactness result holds true with

Navier boundary conditions replaced with Dirichlet boundary conditions.We believe that nontrivial solutions of (17) either with Navier (2) or

Dirichlet (3) boundary conditions do not exist and that the previous lemmamay be strengthened (see Section 3.1 and 3.2). Nevertheless, Lemma 5 issufficient to prove the following compactness property in the second criticalenergy range.


Lemma 6. Let (uh) ⊂ V (Ω) be a Palais–Smale sequence for f at levelc ∈ (S, 24/nS). Then, up to a subsequence, (uh) strongly converges in H2 ∩H1

0 (Ω).

Proof. Assume by contradiction that the sequence (uh) is not relativelycompact in H2 ∩H1

0 (Ω). Then, by Lemma 5 one finds functions u0 ∈ H2 ∩H1

0 (Ω) and u1, ..., uk ∈ D2,2 satisfying (20) and (21). Assume first that allu1, . . . , uk are positive solutions of (14). Then, by Lemma 1 each uj is oftype (11) and attains the best Sobolev constant, i.e.∫

Rn

|uj |2∗ dx =∫

Rn

|∆uj |2 dx = S

[∫Rn

|uj |2∗ dx]2/2∗

, j = 1, . . . k, (22)

which implies∫

Rn |uj |2∗ dx = Sn/4 for j = 1, ..., k. In particular, one obtains

limhf(uh) =

[∫Ω

|u0|2∗ dx+ kSn/4

]4/n

.

If u0 ≡ 0, we get f(uh) → k4/nS, while if u0 6≡ 0 for each k ≥ 1 one has[∫Ω

|u0|2∗ dx+ kSn/4

]4/n

> (k + 1)4/nS.

In any case we have a contradiction with S < limhf(uh) < 24/nS.

We now consider the case in which at least one uj is sign changing or asolution in the half space xn > 0. By Lemmas 2 and 4 we then have forthese uj :∫

Rn

|uj |2∗ dx =∫

Rn

|∆uj |2 dx ≥ 24/nS

[∫Rn

|uj |2∗ dx]2/2∗

, (23)

and hence∫

Rn |uj |2∗ dx ≥ 2Sn/4 , while for the remaining uj , (22) holds

true. In any case, we have limhf(uh) ≥

(2Sn/4

)4/n= 24/nS, again a con-

tradiction.

5. Proof of Theorems 1 and 2

The proof of Theorem 1 is by far more involved than the proof of Theorem 2,because the trivial extension of any function u ∈ H2∩H1

0 (Ω) by 0 does notyield a function in H2(Rn). In particular, for the space H2

0 (Ω) Lemma 7easily follows by this extension argument. A further difference is that thepositivity conclusion of Lemma 3 does not hold in the Dirichlet boundaryvalue case. But in the latter case, one simply has to drop this argument.Therefore we only deal with the proof of Theorem 1.

For all smooth Ω ⊆ Ω let β : V (Ω) → Rn be the “barycenter” map

β(u) =∫

Ω

x|u(x)|2∗ dx.


Since Ω is smooth one finds r > 0 such that Ω is a deformation retract of

Ω+ =x ∈ Rn : d(x, Ω) < r

.

First we show that the energy or, equivalently, the optimal Sobolev con-stants will remain relatively large if we prevent the functions from concen-trating “too close” to their domain of definition.

Lemma 7.

γ = inf

inff(u) : u ∈ V (Ω), β(u) 6∈ Ω+

, Ω smooth subset of Ω

> S.

Proof. Assume by contradiction that for each ε > 0 there exists a smoothΩε ⊆ Ω and uε ∈ H2 ∩H1

0 ∩ C∞(Ωε) such that∫Ωε

|uε|2∗ dx = 1 (24)

∫Ωε

|∆uε|2 dx ≤ S + ε (25)

β(uε) 6∈ Ω+. (26)

Let Uε ∈ H2(Rn) be any entire extension of uε. Since Ωε is smooth, theexistence of such an extension is well known. We emphasize that the quanti-tative properties of Uε outside Ωε (which are expected to blow up for ε 0)will not be used. Further, let 1Ωε

be the characteristic function of Ωε.

Step I. We claim that 1ΩεUε → 0 in L2(Rn) as ε→ 0.

By Theorem 5 there exists C > 0 independent of ε such that

∀u ∈ H2 ∩H10 (Ωε) : ‖∆u‖22 ≥ S‖u‖22∗ +

1C‖u‖21. (27)

Putting together (24), (25) and (27), for each ε > 0 we have

S + ε ≥∫

Ωε

|∆uε|2 dx ≥ S‖uε‖22∗ +1C‖uε‖21 = S +

1C‖1ΩεUε‖21

so that ‖1ΩεUε‖1 → 0 as ε→ 0. By (24) and classical Lp–interpolation, we

also get ‖1ΩεUε‖2 → 0 as ε→ 0.

Step II. The weak* limit (in the sense of measures) of 1Ωε |Uε|2∗ is a Diracmass.

Integration by part shows that ‖∇uε‖2L2(Ωε) ≤ ‖∆uε‖L2(Ωε) ‖uε‖L2(Ωε).Then by Step I and (25), we infer that

‖1Ωε∇Uε‖L2(Rn) = ‖∇uε‖L2(Ωε) → 0.


Therefore, taking any ϕ ∈ C∞c (Rn), as ε→ 0 we find∫

Rn

1Ωε |∆(ϕUε)|2 dx =∫

Rn

1Ωε |ϕ|2|∆Uε|2 dx+ o(1) . (28)

By the L1(Rn) boundedness of the sequences (1Ωε |Uε|2∗) and (1Ωε |∆Uε|2)there exist two bounded non negative measures ν, µ on Rn such that

1Ωε|Uε|2∗ ∗ ν, 1Ωε

|∆Uε|2 ∗ µ (29)

in the sense of measures. By the Sobolev inequality in Ωε we have

∀ε > 0 : S

(∫Rn

1Ωε |Uε|2∗ |ϕ|2∗ dx)2/2∗

≤∫

Rn

1Ωε |∆(ϕUε)|2 dx

and therefore, by (28) and (29), letting ε→ 0 yields

S

(∫Rn

|ϕ|2∗ dν)2/2∗

≤∫

Rn

|ϕ|2 dµ.

By (24) and (25) we also know that

S

(∫Rn

dν

)2/2∗

=∫

Rn

dµ.

Hence by [24, Lemma I.2] there exist x ∈ Ω and σ > 0 such that ν = σδx.From (24) we see that σ = 1.

The contradiction follows by Step II, since β(uε) → x, against (26). Thiscompletes the proof.

According to the previous lemma, we may choose µ such that S < µ <min24/nS, γ. Let ϕ ∈ C∞

c (B1(0)) be such that∫B1(0)

|ϕ|2∗ dx = 1,∫

B1(0)

|∆ϕ|2 dx < µ. (30)

Define for each σ > 0 and y ∈ Rn the function ϕσ,y : Rn → R by setting

ϕσ,y(x) = ϕ

(x− y

σ

)where ϕ is set to zero outside B1(0) . It is readily seen that there existsσ > 0 such that Bσ(y) ⊂ Ω and hence ϕσ,y ∈ C∞

c (Ω) for each y ∈ H . Forall Ω ( Ω let zΩ ∈ D2,2(Rn) be such that zΩ = 1 in Ω\Ω and∫

Rn

|∆zΩ |2 dx = cap(Ω\Ω),

see what follows Definition 1. Note that one has

(1− zΩ)Ty(ϕ) ∈ H2 ∩H10 (Ω) ∀y ∈ H,


where Ty(ϕ) := ϕσ,y

‖ϕσ,y‖2∗. Moreover, for each δ > 0 there exists ε > 0 with

supy∈H

‖zΩTy(ϕ)‖H2∩H10 (Ω) < δ (31)

whenever cap(Ω\Ω) < ε. Then one gets

‖(1− zΩ)Ty(ϕ)‖L2∗ (Ω) 6= 0 ∀y ∈ H,

if ε is sufficiently small. So, if we define the map ΦΩ : H → V (Ω) by

ΦΩ(y) =(1− zΩ)Ty(ϕ)

‖(1− zΩ)Ty(ϕ)‖2∗.

Taking into account (30) and (31), we find ε > 0 such that

supf(ΦΩ(y)) : y ∈ H < µ (32)

provided that cap(Ω\Ω) < ε. From now on ε is fixed subject to the previousrestrictions. Let Ω ⊂ Ω be such that cap(Ω\Ω) < ε and r ∈ (0, r) be suchthat Ω is a deformation retract of

Ω+ =x ∈ Rn : d(x,Ω) < r

.

As in Lemma 7, one obtains

inf f(u) : u ∈ V (Ω), β(u) 6∈ Ω+ > S. (33)

Notice that

inf f(u) : u ∈ V (Ω), β(u) 6∈ Ω+ ≤ sup f(ΦΩ(y)) : y ∈ H (34)

otherwise the map R : H × [0, 1] → Ω given by

R(y, t) = (1− t)y + tβ(ΦΩ(y)) ∀y ∈ H, ∀t ∈ [0, 1]

would deform H in Ω into a subset of Ω+ and then into a subset of Ω (Ω isa deformation retract of Ω+) contradicting the assumptions. Here, in orderto see R(y, t) ∈ Ω, we used the fact that supp ΦΩ(y) ⊂ Bσ(y) and that,since ϕ ≥ 0, β (ΦΩ(y)) ∈ Bσ(y) ⊂ Ω.

Therefore, by combining (32), (33) and (34) one gets

S < inf f(u) : u ∈ V (Ω), β(u) 6∈ Ω+ ≤ sup f(ΦΩ(y)) : y ∈ H (35)

< µ < γ ≤ inff(u) : u ∈ V (Ω), β(u) 6∈ Ω+

.

In what follows we need to find two appropriate levels, such that the corre-sponding sublevel sets

fc = u ∈ V (Ω) : f(u) ≤ c


cannot be deformed into each other. For this purpose let c1, c2 > S be suchthat

c1 < inff(u) : u ∈ V (Ω), β(u) 6∈ Ω+ ,

c2 = supf(ΦΩ(y)) : y ∈ H.

Assume by contradiction that there exists a deformation ϑ of fc2 into fc1 ,i.e.

ϑ : fc2 × [0, 1] → fc2 , ϑ( . , 0) = Idfc2 and ϑ(fc2 , 1) ⊆ fc1 .

We define H : H × [0, 1] → Ω by setting for each x ∈ H

H (x, t) =

(1− 3t)x+ 3tβ(ΦΩ(x)) if t ∈ [0, 1/3]%(β(ϑ(ΦΩ(x), 3t− 1))) if t ∈ [1/3, 2/3]% (% (β (ϑ(ΦΩ(x), 1)) , 3t− 2)) if t ∈ [2/3, 1]

where % : Ω+ → Ω is a retraction and % : Ω+ × [0, 1] → Ω+ is a continuousmap with %(x, 0) = x and %(x, 1) ∈ Ω for all x ∈ Ω+. In order to see thatH (x, t) ∈ Ω, one should observe that c2 < γ and ϑ(ΦΩ(x), 3t − 1) ∈ fc2 ,hence β(ϑ(ΦΩ(x), 3t− 1)) ∈ Ω+.

As ϑ(ΦΩ(x), 1) ∈ fc1 for each x ∈ H and

c1 < inf f(u) : u ∈ V (Ω), β(u) 6∈ Ω+ ,

then for each x ∈ Hβ(ϑ(ΦΩ(x), 1)) ∈ Ω+

and H is a deformation of H in Ω into a subset of Ω, in contradiction withour assumptions. Then the sublevel set

fc2 = u ∈ V (Ω) : f(u) ≤ c2

cannot be deformed into

fc1 = u ∈ V (Ω) : f(u) ≤ c1 .

Hence, by combining Lemma 6 with the standard deformation lemma (see[31, Theorem 4.6] and also [10, Lemma 27.2], [38, Theorem 3.11]) one obtainsa constrained critical point uΩ such that

S < f(uΩ) ≤ supf (ΦΩ(y)) : y ∈ H < µ < 24/nS.

Finally, uΩ does not change sign by Lemma 3.


6. Proof of Theorems 3, 4 and 5

Proof of Theorem 3. We assume that Ω = B = B1(0) is the unit balland that (8) with λ > 0 has a positive solution u. As we have alreadymentioned above, the result by Troy [41, Theorem 1] shows that u is radiallysymmetric. As the right hand side in (8) is positive, an iterated applicationof the maximum principle for −∆ yields that −∆u and u are strictly positiveand strictly radially decreasing. We will exploit the following Pohozaev typeidentity, which may be taken e.g. from [42, Lemma 3.9]:

2λ∫

B

u2 dx =∫

∂B

(∂u

∂r

)(∂

∂r(−∆u)

)dS. (36)

As u is radially symmetric we may proceed as follows with the term on theright hand side; here, en denotes the n-dimensional volume of the unit ballB and c(n) denotes a generic positive constant which may vary from line toline: ∫

∂B

(∂u

∂r

)(∂

∂r(−∆u)

)dS = −n en

∂u

∂r(1) · ∂

∂r(∆u)(1)

=1n en

(∫∂B

(−∂u∂r

)dS

)(∫∂B

(∂

∂r(∆u)

)dS

)=

1n en

(∫B

(−∆u) dx)(∫

B

∆2u dx

). (37)

The first integral needs some further consideration. First of all, by makinguse of homogeneous Navier boundary conditions, we find:∫

B

(1− |x|2

)∆2u dx = −

n∑j=1

∫B

(−2xj)(∆u)xjdx = 2n

∫B

(−∆u) dx.

Furthermore, as we have already mentioned, u and by (8) also ∆2u arepositive and strictly radially decreasing. This fact can be used to get rid ofthe degenerate weight factor 1− |x|2 in the previous term:∫

B

∆2u dx ≤∫|x|≤1/2

∆2u dx+∫

1/2≤|x|≤1

∆2u dx

≤∫|x|≤1/2

∆2u dx+ c(n)∆2u

(12

)≤∫|x|≤1/2

∆2u dx+ c(n)∫|x|≤1/2

∆2u dx

≤ c(n)∫

B

(1− |x|2

)∆2u dx.

Combining this estimate with the previous identity, we obtain from (36)and (37):

λ

∫B

u2 dx ≥ c(n)(∫

B

∆2u dx

)2

= c(n)‖∆2u‖21. (38)


By a duality argument and using elliptic estimates, we find

‖u‖2 ≤ c(n)‖∆2u‖1, provided n < 8. (39)

Observe that L2−estimates for the biharmonic operator under homogeneousNavier boundary conditions follow immediately from the L2−estimates forthe Laplacian.

Combining (38) and (39), we see that for any positive radial solution uof (8), we have

λ

∫B

u2 dx ≥ c(n)∫

B

u2 dx;

where the constant c(n) is in particular independent of u. As u > 0, we havenecessarily λ ≥ c(n).

Proof of Theorem 4. Assume by contradiction that u is a nontrivial radialsolution of (9). According to the remarks in Section 3.2 on the nonexistenceof positive solutions, we may assume that u is sign changing. Further, ucannot have infinitely many oscillations near ∂B, because in this case, asu ∈ C4(B) we would also have ∆u = 0 as well as d

dr∆u = 0 on ∂B and couldextend u by 0 to the whole of Rn as a solution of the differential equation.The unique continuation property [33] would give the desired contradiction.

As one may replace u with −u if necessary, we may thus assume thatthere exists a number a ∈ (0, 1) such that

u(x) = 0 for |x| = a, u(x) > 0 for a < |x| < 1;

moreover, by a Pohozaev-type identity,

u(x) = ∇u(x) = ∇2u(x) = 0 for |x| = 1. (40)

Clearly ddru(a) ≥ 0. We define

f(x) := |u(x)|2∗−2u(x)

and compare u with the solution v of∆2v(x) = f(x) for a < |x| < 1,

v(x) = ∇v(x) = 0 for |x| ∈ a, 1.

Obviously v > 0 on a < |x| < 1. We now make use of comparison results,which hold true for radial solutions of biharmonic inequalities in the annulusx : a < |x| < 1 and can e.g. be found in [8, Section 2]. First one obtains

d2

dr2v(1) > 0. (41)

Next, as u and v satisfy the same differential equation, if ddru(a) = 0 it

follows that u ≡ v while if ddru(a) > 0 = d

drv(a) we find u > v in a < |x| < 1.In both cases (41) gives

d2

dr2u(1) > 0


and hence a contradiction with (40).

Proof of Theorem 5. We keep p ∈ [1, nn−4 ) fixed as in the theorem.

Let B = B1(0) denote the unit ball, centered at the origin. By scaling andTalenti’s comparison principle [40, Theorem 1] it suffices to prove that thereexists CB > 0 with

‖∆u‖22 ≥ S‖u‖22∗ +1CB

‖u‖2p (42)

for each u ∈ H2∩H10 (B), which is radially symmetric and radially decreasing

with respect to the origin (and hence positive). Let us set

Sλ,p = infu∈H2∩H1

0(B)u radially decreasing

‖∆u‖22 − λ‖u‖2p‖u‖22∗

.

If by contradiction (42) does not hold, one gets

∀λ > 0 : Sλ,p < S. (43)

Let us fix some λ ∈ (0, λ1,p), where

λ1,p := infu∈H2∩H1

0 (B)\0

‖∆u‖22‖u‖2p

(44)

is the first positive eigenvalue of∆2u = λ‖u‖2−p

p |u|p−2u in Bu = ∆u = 0 on ∂B.

Arguing along the lines of [6] and [15], one finds a smooth positive radialstrictly decreasing solution of

∆2u = u(n+4)/(n−4) + λ‖u‖2−pp |u|p−2u in B

u = ∆u = 0 on ∂B.(45)

On the other hand, for any positive radially decreasing solution of (45) onehas the following variant of Pohozaev’s identity:

λ

(2− n(p− 2)

2p

)‖u‖2p =

∫∂B

(∂u

∂r

)(∂

∂r(−∆u)

)dS. (46)

As in the proof of Theorem 3, we conclude

λ‖u‖2p ≥ c(n, p)‖∆2u‖21 ≥ c(n, p) ‖u‖2p, (47)

as we have assumed that p < n/(n − 4). For λ > 0 sufficiently small, weobtain a contradiction.


7. Appendix: Proof of Lemma 5

We prove the corresponding statement for the “free” functional

EΩ(u) =12

∫Ω

|∆u|2 dx− 12∗

∫Ω

|u|2∗ dx

which is defined on the whole space H2 ∩H10 (Ω). More precisely, we have

Lemma 8. Let (uh) ⊂ H2 ∩ H10 (Ω) be a Palais–Smale sequence for EΩ

at level c ∈ R. Then either (uh) is relatively compact in H2 ∩ H10 (Ω) or

there exist k > 0 nonzero functions uj ∈ D2,2(Ω0,j), j = 1, . . . , k, withΩ0,j either the whole Rn or a half space, solving either (14) or (17) withboundary condition (2) and a solution u0 ∈ H2 ∩ H1

0 (Ω) of (5) such that,as h→∞:

uh u0 in H2 ∩H10 (Ω),

‖∆uh‖22 → ‖∆u0‖22 +k∑

j=1

‖∆uj‖22, EΩ(uh) → EΩ(u0) +k∑

j=1

EΩ0,j(uj).

In the second order situation, the corresponding result is due Struwe[37]. Related generalisations of the Struwe compactness lemma to higherorder problems can be found in [1,3,11,19]. However, none of these appliesdirectly to our situation. A particular difficulty here arises from the existenceof the boundary ∂Ω in combination with Navier boundary conditions.

Proof of Lemma 8.Step I. Reduction to the case uh 0.

By well known arguments [6] we know that there exists u ∈ H2∩H10 (Ω)

such that uh u (up to a subsequence) and E′Ω(u) = 0, namely u solves

(5). Moreover, by the Brezis–Lieb lemma we infer that (uh − u) is again aPalais–Smale sequence for E. Therefore we may assume that

uh 0, E(uh) → c ≥ 2nSn/4. (48)

Indeed, if c < 2nS

n/4, then we are in the compactness range of E and byarguing as in [6], uh → 0 up to a subsequence. In this case the statementfollows with k = 0.

Assuming (48) and arguing as for the proof of [37, (3.2), p.187] we deduce∫Ω

|∆uh|2 dx ≥ Sn/4 + o(1). (49)

Let L ∈ N be such that B2(0) is covered by L balls of radius 1. By continuityof the maps

R 7→ supy∈Ω

∫BR−1 (y)∩Ω

|∆uh|2 dx, y 7→∫

BR−1 (y)∩Ω

|∆uh|2 dx


and (49), for h large enough one finds Rh > 2/diam(Ω) and xh ∈ Ω suchthat∫

BR−1h

(xh)∩Ω

|∆uh|2 dx = supy∈Ω

∫B

R−1h

(y)∩Ω

|∆uh|2 dx =1

2LSn/4. (50)

When passing to a suitable subsequence, three cases may now occur:Case I. Rh +∞ and (Rhd(xh, ∂Ω)) is bounded;Case II. (Rhd(xh, ∂Ω)) → +∞;Case III. (Rh) is bounded.

Step II. Preliminaries for Case I.For every x ∈ Rn let us denote by x′ its projection onto Rn−1, so that

x = (x′, xn). Since d(xh, ∂Ω) → 0, up to a subsequence xh → x0 ∈ ∂Ω and%h := Rhd(xh, ∂Ω) → %. Moreover, for sufficiently large h (say h ≥ h) thereexists a unique yh ∈ ∂Ω such that d(xh, ∂Ω) = |yh − xh|.

For all h ≥ h, up to a rotation and a translation, we may assume thatyh = 0 and that the tangent hyperplane H to ∂Ω at 0 has equation xn = 0so that xh−yh ⊥ H. Then, by the smoothness of ∂Ω, there exist an (n−1)–dimensional ball Bσ(0) of radius σ > 0 (independent of h) and smooth mapsψh : Bσ(0) → R such that in local orthonormal coordinate systems over thetangent hyperplanes at yh, we have:

∂Ω ∩(Bσ(0)× [−σ, σ]

)=(x′, ψh(x′)) : x′ ∈ Bσ(0)

. (51)

Furthermore, there exists C > 0 (independent of h) such that

|ψh(x′)| ≤ C|x′|2, |∇ψh(x′)| ≤ C|x′|, |D2ψh(x′)| ≤ C ∀x′ ∈ Bσ(0).(52)

We now rescale the domain Ω by setting

Ωh = Rh

(Ω − xh

),

so that xh is mapped into the origin 0 while the origin is mapped into(0,−%h) and (51) becomes

∂Ωh ∩(BσRh

(0)× [−σRh − %h, σRh − %h])

=

=(

x′, Rhψh

(x′

Rh

)− %h

): x′ ∈ BσRh

(0).

Let us set

ωh =

(x′, x) ∈ Rn : x′ ∈ BRhσ(0),

Rhψh

(x′

Rh

)− %h < xn < Rhψh

(x′

Rh

)− %h +Rhδ


where δ > 0 and sufficiently small to have ωh ⊂ Ωh. Consider the injectivemap

χh : BRhσ(0)× [−Rhδ,Rhδ] → Rn

(x′, xn) 7→(x′, Rhψh

(x′

Rh

)+ xn − %h

)and its inverse χ−1

h . We observe that χh

(BRhσ(0)× [0, Rhδ]

)= ωh and

that χh is bijective on these sets. Thanks to (52), after some computations,one sees that (χh) converges in C2

loc(xn ≥ 0) to the translation by thevector (0,−%). Therefore,

Ω0 = (x′, xn) ∈ Rn : xn > −%is the local uniform limit of (Ωh). In particular, for every ϕ ∈ C∞

c (Ω0) wealso have that ϕ ∈ C∞

c (Ωh) for sufficiently large h. This will be used below.Let us now set

vh(x) = R4−n

2h uh

(xh +

x

Rh

), (53)

so that vh ∈ H2 ∩ H10 (Ωh). By boundedness of (uh) we infer that there

exists C > 0 such that ‖vh‖H2(Ωh) ≤ C. Let 1Ωhdenote the characteristic

function of Ωh, then the sequence (1Ωhvh) is bounded in D1,2n/(n−2)(Rn)

(and in L2∗(Rn)) so that, up to a subsequence, we have

1Ωhvh v0 in D1,2n/(n−2) ∩ L2∗(Rn) (54)

where supp(v0) ⊆ Ω0 and v0|xn=−% = 0. Moreover, as (1ΩhD2vh) is bounded

in L2(Rn), by weak continuity of distributional derivatives, we deduce∫Ωh

D2ijvh ϕdx→

∫Ω0

D2ijv0 ϕdx

for all ϕ ∈ C∞c (Ω0) and i, j = 1, . . . , n. In particular, v0 has in Ω0 second

order weak derivatives.

Step III. The limiting function v0 in (54) solves (5) in Ω0.Fix ϕ ∈ C∞

c (Ω0), then for h large enough we have supp(ϕ) ⊂ Ωh. Defineϕh ∈ C∞

c (Ω) by setting

ϕh(x) = Rn−4

2h ϕ

(Rh(x− xh)

).

Therefore, (D2ϕh) being bounded in L2(Ω) and taking into account (53),one obtains

o(1) =E′Ω(uh)(ϕh) = R

n2h

∫Ω

∆uh∆ϕ(Rh(x− xh)

)dx

−Rn−4

2h

∫Ω

|uh|2∗−2uhϕ(Rh(x− xh)

)dx

=∫

Rn

∆vh∆ϕdx−∫

Rn

|vh|2∗−2vhϕdx

=∫

Rn

∆v0∆ϕdx−∫

Rn

|v0|2∗−2v0ϕdx+ o(1).


Then v0 ∈ D2,2(Ω0) solves (5) in distributional sense. The delicate point isto see that ∆v0 = 0 on ∂Ω0. To this end, let ϕ ∈ C2

c (Rn) with ϕ = 0 on∂Ω0 and define

ϕh(x) = ϕ(χ−1

h (x)− (0, %)).

Notice that for h large enough, also ϕh ∈ C2c (Rn) with ϕh = 0 on ∂Ωh.

Taking into account that (χ−1h ) tends to the translation by (0, %), we obtain

ϕh → ϕ in C2(Rn) and∫Ω0

(∆v0∆ϕ − |v0|2∗−2v0ϕ) dx = limh

∫Ωh

(∆vh∆ϕh − |vh|2∗−2vhϕh) dx

= limh

∫Ω

[∆uh∆

(R

n−42

h ϕh(Rh(x− xh)))−

− |uh|2∗−2uhRn−4

2h ϕh(Rh(x− xh))

]dx

= limhE′

Ω(uh)(R

n−42

h ϕh(Rh(x− xh)))

= 0.

Therefore, by extending any given ϕ ∈ H10 ∩H2 (Ω0) oddly with respect to

xn−% as a function of H2(Rn) and then by approximating it by a sequenceof C2 functions (ϕk) with ϕk = 0 on ∂Ω0, we get

∀ϕ ∈ H10 ∩H2(Ω0) :

∫Ω0

∆v0∆ϕdx =∫

Ω0

|v0|2∗−2v0ϕdx,

which, as pointed out in [4, p. 221] implies that v0 is also a strong solutionof (5) in Ω0.

Step IV. The limiting function v0 in (54) is nontrivial.Let ϕ ∈ C∞

c (Rn) such that Ω0 ∩ suppϕ 6= ∅; then for h large enough, wemay define

v0,h : Ωh ∩ supp(ϕ) → R, v0,h(x) = v0(χ−1

h (x)− (0, %)). (55)

Since v0 ∈ C2(Ω0), v0 = ∆v0 = 0 on ∂Ω0 and (χh) converges to thetranslation by the vector (0,−%), we get as h→ +∞

‖1Ω0v0 − 1Ωhv0,h‖L2(supp(ϕ)) = o(1),

‖1Ω0v0 − 1Ωhv0,h‖L2∗ (supp(ϕ)) = o(1)

‖1Ω0∇v0 − 1Ωh∇v0,h‖L2(supp(ϕ)) = o(1),

‖1Ω0∆v0 − 1Ωh∆v0,h‖L2(supp(ϕ)) = o(1).

(56)

To symplify the notations, in what follows we omit the 1Ωhin front of

vh,∇vh,∆vh and 1Ω0 in front of v0,∇v0,∆v0. Then by (56) and some com-


putations, one obtains, as h→ +∞∫Rn

|∆(ϕv0 − ϕvh)|2 dx =∫

Ωh∩supp(ϕ)

|∆(ϕv0,h − ϕvh)|2 dx+ o(1)

≥ S

(∫Ωh∩supp(ϕ)

|ϕ(v0,h − v0)|2∗ dx

)2/2∗

+ o(1)

≥ S

(∫Rn

|ϕ(v0 − vh)|2∗ dx)2/2∗

+ o(1).

By compact embedding one has vh → v0 in L2loc(Rn) and thanks to an

integration by parts one gets

‖∇(ϕ(vh − v0))‖2 ≤ ‖ϕ(vh − v0)‖2‖∆(ϕ(vh − v0))‖2

and therefore ∇vh → ∇v0 in L2loc(Rn). Hence the previous inequality yields∫

Rn

ϕ2|∆(v0 − vh)|2 dx ≥ S

(∫Rn

|ϕ|2∗ |v0 − vh|2∗ dx)2/2∗

+ o(1),

which implies ∫Rn

ϕ2 dµ ≥ S

(∫Rn

|ϕ|2∗ dν)2/2∗

,

where dµ and dν denote respectively the weak∗ limits of |∆(v0 − vh)|2 and|v0 − vh|2∗ in the sense of measures.

The rest of this step – to show v0 6≡ 0 – is now very closely along the linesof [1, Lemma 2] and [38, p. 173]. We emphasize that here, the normalizationcondition (50) is exploited.

Step V. In Case II, a solution of (14) appears.This follows by arguing as in [1], see also [37]. After rescaling as in (53),

(vh) converges to a solution of (14).

Step VI. Case III cannot occur.By contradiction, assume that (Rh) is bounded. Then, being Rh >

2/diam(Ω), we may assume that, up to a subsequence,

xh → x0 ∈ Ω, Rh → R0 > 0.

Let us set Ω0 = R0(Ω − x0) and

vh = R4−n

20 uh

(x0 +

x

R0

).

As in Case I (with the obvious simplifications), one gets for a subsequencevh v0 ∈ H2∩H1

0 (Ω0) and that v0 6≡ 0 solves (5) in Ω0. But this is absurdsince uh 0.


Step VII. Conclusion.If (uh) is a Palais–Smale sequence for EΩ , then by Step I its weak limit

u0 solves (5). By Steps III, IV and V the “remaining part” (uh−u0) suitablyrescaled gives rise to a nontrivial solution v0 of (14) (if Case II occurs) or (17)(if Case I occurs). With help of this solution, we construct from (uh− u0) anew Palais–Smale sequence (wh) for EΩ in H2 ∩H1

0 (Ω) at a strictly lowerenergy level. In the case where we find the solution v0 in the whole space,one proceeds precisely as in [37], cf. also [1]. In the case that v0 is a solutionof (17) in a half space Ω0 we again have to use the locally deformed versionsv0,h in Ωh of v0. These have been defined in (55). Let ϕ ∈ C∞

c (Rn) be anycut–off function with 0 ≤ ϕ ≤ 1, ϕ = 1 in B1(0) and ϕ = 0 outside B2(0).We put

wh(x) := (uh − u0)(x)−R(n−4)/2h v0,h

(Rh(x− xh)

)ϕ(√

Rh(x− xh)).

First we remark that wh is indeed well defined as ϕ = 0 for Rh|x − xh| ≥2√Rh and as the domain of definition of v0,h grows at rate Rh. Further we

notice that χh and χ−1h converge uniformly to translations even on BRh

(0).For this reason we have

v0,h(·)ϕ(

·√Rh

)→ v0 in D2,2(Ω0)

and also in D1,2n/(n−2)(Ω0) and in L2∗(Ω0). Hence, we have

EΩ(wh) = EΩ(uh)− EΩ(u0)− EΩ0(v0) + o(1)

and E′Ω(wh) → 0 strongly in

(H2 ∩H1

0 (Ω))∗. Now, the same procedure

from Steps II to VI is applied to (wh) instead of (uh − u0). As v0 6≡ 0 andhence

EΩ0(v0) ≥2nSn/4,

this procedure stops after finitely many iterations.

Acknowledgement. The authors are grateful to Nicola Garofalo for raisingtheir attention on the paper [22] and to the referees for a very careful readingof the manuscript.

References

1. C.O. Alves, J.M. do O, Positive solutions of a fourth-order semilinear probleminvolving critical growth, preprint, 2001.

2. A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the criticalSobolev exponent: the effect of the topology of the domain, Comm. Pure Appl.Math. 41, 1988, 253–294.

3. T. Bartsch, T. Weth, M. Willem, A Sobolev inequality with remainder termand critical equations on domains with nontrivial topology for the polyhar-monic operator, preprint, 2002.


4. F. Bernis, J. Garcıa Azorero, I. Peral, Existence and multiplicity of nontrivialsolutions in semilinear critical problems of fourth order, Adv. DifferentialEquations 1, 1996, 219–240.

5. T. Boggio, Sulle funzioni di Green d’ordine m, Rend. Circ. Mat. Palermo 20,1905, 97–135.

6. H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations in-volving critical Sobolev exponents, Comm. Pure Appl. Math. 36, 1983, 437–477.

7. J.–M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad.Sci., Paris, Sr. I 299, 1984, 209-212.

8. R. Dalmasso, Elliptic equations of order 2m in annular domains, Trans. Amer.Math. Soc. 347, 1995, 3575–3585.

9. E.N. Dancer, A note on an equation with critical exponent, Bull. LondonMath. Soc. 20, 1988, 600–602.

10. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin etc. 1985.11. F. Ebobisse, M.O. Ahmedou, On a nonlinear fourth order elliptic equation

involving the critical Sobolev exponent, preprint, 2001.12. D.E. Edmunds, D. Fortunato, E. Jannelli, Critical exponents, critical dimen-

sions and the biharmonic operator, Arch. Ration. Mech. Anal. 112, 1990,269–289.

13. F. Gazzola, Critical growth problems for polyharmonic operators, Proc. RoyalSoc. Edinburgh Sect. A 128, 1998, 251–263.

14. F. Gazzola, H.–Ch. Grunau, Critical dimensions and higher order Sobolevinequalities with remainder terms, NoDEA Nonlinear Differential EquationsAppl. 8, 2001, 35–44.

15. H.–Ch. Grunau, Positive solutions to semilinear polyharmonic Dirichlet prob-lems involving critical Sobolev exponents, Calc. Var. Partial Differential Equa-tions 3, 1995, 243–252.

16. H.–Ch. Grunau, Polyharmonische Dirichletprobleme: Positivitat, kritische Ex-ponenten und kritische Dimensionen, Habilitationsschrift, Universitat Bay-reuth 1996.

17. H.–Ch. Grunau, On a conjecture of P. Pucci and J. Serrin, Analysis 16, 1996,399–403.

18. H.–Ch. Grunau, G. Sweers, Positivity for perturbations of polyharmonic op-erators with Dirichlet boundary conditions in two dimensions, Math. Nachr.179, 1996, 89-102.

19. E. Hebey, F. Robert, Coercivity and Struwe’s compactness for Paneitz typeoperators with constant coefficients, Calc. Var. Partial Differential Equations13, 2001, 491–517.

20. E. Jannelli, The role played by space dimension in elliptic critical problems,J. Differential Equations 156, 1999, 407–426.

21. B. Kawohl, G. Sweers, On ’anti’-eigenvalues for elliptic systems and a questionof McKenna and Walter, Indiana U. Math. J., to appear.

22. C. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequal-ities and unique continuation, in: Harmonic analysis and partial differentialequations, (El Escorial, 1987), 69–90, Lect. Notes Math. 1384, Springer, 1989.

23. C.S. Lin, A classification of solutions of a conformally invariant fourth orderequation in Rn, Comment. Math. Helv. 73, 1998, 206–231.

24. P.L. Lions, The concentration–compactness principle in the calculus of varia-tions. The limit case. I, Rev. Mat. Iberoamericana 1, 1985, 145–201.

25. S. Luckhaus, Existence and regularity of weak solutions to the Dirichlet prob-lem for semilinear elliptic systems of higher order, J. Reine Angew. Math.306, 1979, 192–207.

26. E. Mitidieri, On the definition of critical dimension, unpublished manuscript,1993.

28 Filippo Gazzola et al.: Biharmonic equations at critical growth

27. E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differ-ential Equations 18, 1993, 125–151.

28. E. Mitidieri, G. Sweers, Weakly coupled elliptic systems and positivity, Math.Nachr. 173, 1995, 259-286.

29. J.J. Moreau, Decomposition orthogonale d’un espace hilbertien selon deuxcones mutuellement polaires, C. R. Acad. Sci. Paris 255, 1962, 238–240.

30. P. Oswald, On a priori estimates for positive solutions of a semilinear bihar-monic equation in a ball, Comment. Math. Univ. Carolinae 26, 1985, 565–577.

31. R. Palais, Critical point theory and the minimax principle, Proc. Symp. PureMath. 15, 1970, 185–212.

32. D. Passaseo, The effect of the domain shape on the existence of positive

solutions of the equation ∆u + u2∗−1 = 0, Topol. Meth. Nonlinear Anal. 3,1994, 27–54.

33. R.N. Pederson, On the unique continuation theorem for certain second andfourth order elliptic equations, Comm. Pure Appl. Math. 11, 1958, 67–80.

34. S.I. Pohozaev, Eigenfunctions of the equation ∆u + λf(u) = 0, Soviet. Math.Dokl. 6, 1965, 1408–1411.

35. P. Pucci, J. Serrin, A general variational identity, Indiana Univ. Math. J. 35,1986, 681–703.

36. P. Pucci, J. Serrin, Critical exponents and critical dimensions for polyhar-monic operators, J. Math. Pures Appl. 69, 1990, 55–83.

37. M. Struwe, A global compactness result for elliptic boundary value problemsinvolving limiting nonlinearities, Math Z. 187, 1984, 511–517.

38. M. Struwe, Variational methods, Springer, Berlin etc. 1990.39. C.A. Swanson, The best Sobolev constant, Applicable Anal. 47, 1992, 227–

239.40. G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup.

Pisa Cl. Sci. (IV) 3, 1976, 697–718.41. W.C. Troy, Symmetry properties in systems of semilinear elliptic equations,

J. Differential Equations 42, 1981, 400–413.42. R.C.A.M. van der Vorst, Variational identities and applications to differential

systems, Arch. Ration. Mech. Anal. 116, 1991, 375–398.43. R.C.A.M. van der Vorst, Best constant for the embedding of the space H2 ∩

H10 (Ω) into L2N/(N−4)(Ω), Differential Integral Equations 6, 1993, 259–276.

44. R.C.A.M. van der Vorst, Fourth order elliptic equations with critical growth,C. R. Acad. Sci. Paris, Serie I 320, 1995, 295–299.

existence and nonexistence results for critical growth

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